Localization in sensor networks - A matrix regression approach

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Localization in sensor networks - A matrix regression

approach

Paul Honeine, Cédric Richard, Mehdi Essoloh, Hichem Snoussi

To cite this version:

LOCALIZATION IN SENSOR NETWORKS – A MATRIX REGRESSION APPROACH

Paul Honeine, C´edric Richard, Mehdi Essoloh, Hichem Snoussi

Institut Charles Delaunay (FRE CNRS 2848)- LM2S - Universit´e de technologie de Troyes

12 rue Marie Curie, BP 2060, 10010 Troyes cedex, France - fax. +33.3.25.71.56.99

ABSTRACT

In this paper, we propose a new approach to sensor local-ization problems, based on recent developments in machine leaning. The main idea behind it is to consider a matrix regression method between the ranging matrix and the ma-trix of inner products between positions of sensors, in order to complete the latter. Once we have learnt this regression from information between sensors of known positions (bea-cons), we apply it to sensors of unknown positions. Re-trieving the estimated positions of the latter can be done by solving a linear system. We propose a distributed algo-rithm, where each sensor positions itself with information available from its nearby beacons. The proposed method is validated by experimentations.

1. INTRODUCTION

In ad-hoc wireless sensor networks, a large number of ap-plications require location awareness of the sensors, includ-ing trackinclud-ing, environmental monitorinclud-ing and many military applications. Without the knowledge of its position, the in-formation captured by a sensor becomes obsolete. The main building block of these networks is a low-cost sensor, with low power resources, leaving no room to (absolute) self-positioning device. To overcome this drawback, one in-cludes in the network a small number of sensors with known positions (and sometimes high power and communication capabilities). These sensors, often known by anchors or beacons and designated hereafter by the latter, communicate to other sensors information allowing the latter to estimate their positions. For this purpose, each sensor determines ranging (distance) measurements with other sensors, from some measurements such as the received signal strength in-dication (RSSI), the connectivity, the hop count, the time difference of arrival, ... Most work on localization in sensor networks considers either multidimensional scaling (MDS) techniques or semidefinite programming (see [1, 2], and ref-erences therein), in order to determine a function that links the ranging of the sensors to their positions, based on the known positions of some beacons.

{FirstName.SecondName}@utt.fr

Introduced by Aronszajn in the mid 50s, the usefulness of reproducing kernels has been demonstrated in the last 15 years in the field of pattern recognition with the statistical learning theory and the so called kernel machines, such as support vector machines (SVM) and kernel principal com-ponent analysis (kernel-PCA) [3]. Reproducing kernels pro-vide new insights into sensor networks research field. This has been known for a while, as many researchers in sen-sor networks focus on detection, tracking and classification using kernel machines. In recent years, there has been an increasing interest in this framework for localizing sensors, with kernel-PCA [4, 5], SVM [6], least squares regression [7], and manifold regularization [8].

2. THE MATRIX REGRESSION METHOD

Consider a network of m sensors of unknown positions

andn beacons of known positions, living in a d-dimension

(2D or 3D) space. Let X and Y be the coordinate ma-trices of beacons and sensors, respectively, of size n-by-d ann-by-d m-by-n-by-d, ann-by-d [X⊤ _Y⊤_]⊤ _{the overall coordinate}

ma-trix. The inner product between their coordinates is given by

P=X Y



[X⊤_Y⊤_{], which can be decomposed into four block}

submatrices, Px = XX⊤, Pyx = YX⊤, Pxy = P⊤yx, and

Py= YY⊤, as illustrated in (1) with unknown submatrices

set to gray-color. On the other hand, we have the overall ranging matrix, denoted by K with entriesκ(xi, xj),

sim-ilarly decomposed into Kx, Kyx, Kxy, and Ky, as given

in (1). Kx Kxy Kyx Ky | {z } K

→

In a conventional regression problem, one seeks a func-tion φ(·) that links an input variable x into a response

(output) variablez, under the constraints φ(xi) = zi for

all available training data{(x1, z1), . . . , (xn, zn)}. While

there exists an infinite number of functions verifying such constraints, one considers functions with some regularizing properties (such as smoothness). This can be done by re-stricting the hypothesis space to the RKHS of a given repro-ducing kernel, sayκ(·, ·). Moreover, from the Representer

Theorem [10], the optimal function has the form

φ(·) =

n

X

k=1

αkκ(xk, ·). (2)

For instance, this is true for kernel-PCA, where each princi-pal axisφ(·) is determined by its n coefficients αk, obtained

by an eigen-decomposition of then-by-n matrix of entities κ(xi, xj), thus Kx. Sinceφ(x) corresponds to the

princi-pal coordinate of x, the latter can be represented into a low-dimensional space by considering only a couple of princi-pal coordinates. Since this is the essence of both MDS and kernel-PCA techniques, localization in sensor networks us-ing kernel-PCA is proposed in [4], or more recently [5] (see [11] for a connection to MDS).

In what follows, we consider the general case of deter-mining a set of optimal functions, fully described by their coefficients, which identifies the mapping between the two matrices described in (1). This is known as a matrix re-gression problem [9], between the input dataκ(xi, xj) of

matrix K and the output xix⊤j of P. We learn this problem

from the available input-output couples, i.e. Kx and Px.

For this, we consider a model of the form

Ψ(xi, xj) = xix⊤j, (3)

and determine it from inter-beacon information. As above, we consider a particular form of Ψ, by

rep-resenting each x in some coordinates obtained with a set of functions, [φ1(x) φ2(x) · · · ]⊤, leading to

Ψ(xi, xj) = [φ1(xi) φ2(xi) · · · ][φ1(xj) φ2(xj) · · · ]⊤ =

P

hφh(xi)φh(xj). In analogy to kernel-PCA where these

are principal coordinates obtained from principal axes con-structed from available data κ(xk, xℓ), we consider the

same form as (2) for allφh’s, constructed from Kx.

There-fore one should determine for eachφh the optimal

coeffi-cient vector α= [α1α2 · · · αn]⊤, with

φh(xi) φh(xj) = _Xn k=1 αkκ(xk, xi) _Xn ℓ=1 αℓκ(xℓ, xj)  = κ⊤i α α⊤κj

where κiis thei-th column vector of Kx. From the sum of

these terms and (3), we get xix⊤j = Ψ(xi, xj) = κ⊤i A κj,

where A is a coefficient matrix. Since this should be satis-fied for all beacons, i.e. i, j = 1, . . . , n, we can write the

optimization problem in matrix form, with

min

A kPx− K ⊤

xA Kxk2F, (4)

wherek · kF is Frobenius norm. Once we obtain the

opti-mal coefficient matrix A, we can apply the resulting map to sensors with unknown positions, with xiy⊤j = Ψ(xi, yj).

From expressions above, we obtain the matrix expression

Pxy= K⊤xA Kxy. (5)

The optimization problem (4)-(5) can be solved by writing (4) as a generalized eigen-decomposition problem and in-jecting the resulting matrix in (5). As we notice that both expressions contain the matrix T = K⊤

xA, we propose to

solve the following equivalent optimization problem :

min

T kPx− T Kxk 2

F, and Pxy= T Kxy. (6)

This is a linear problem yielding T = PxK−1x , and thus

Pxy = PxK−1x Kxy. (7)

3. SENSOR POSITION ESTIMATION

After estimating the matrix Pxy of inner products of

of the Nystr¨om method, initially developed in the machine learning community to approximate a matrix by another ma-trix of lower rank [12]. In our case, on the one hand we have by construction Px = XX⊤, thus ad-rank matrix. From

its eigen-decomposition, we have

Px= UdΛdU⊤d,

where Λd is a diagonal matrix of thed nonzero

eigenval-ues of Px, and Udthe matrix whose columns are the

corre-sponding eigenvectors. By identification, we get

X= Ud(Λd)1/2 (8)

On the other hand, we can write Pxy = XY⊤ where Y is

the coordinate matrix to be identified. By injecting (8) into this definition, we get the coordinates of them sensors from

Y⊤_{= (Λ}

d)−1/2U⊤d Pxy. (9)

Since the resulting coordinates are determined in the space defined by the eigenvectors, one must conduct a final step of mapping them, with a linear (or affine to be more precise) transformation, into the initial space of beacons. Such step is commun to MDS techniques.

4. DISTRIBUTED ALGORITHM

In section 2, we considered completing the inner-product matrix by inverting the ranging matrix of the beacons. This stipulates that the beacons communicate to each other, in a peer-to-peer fashion or a multi-hop strategy. However, in practice, beacons may not be in the range of each other. Moreover, the matrices may become too cumbersome to in-vert and manipulate for a large scale sensor network. To overcome this problem, we propose a distributed algorithm, where each sensor gets information from nearby beacons in order to find its own position. In other words, any sensori

defines a set of neighboring1beacons with a submatrix of

Kxand its counterpart in Px, denoted respectively Kiand

Pi, and Xithe corresponding coordinates. While the global

optimization problem (6) leads to expression (7), by consid-ering the distributed approach, we get

pi= PiK−1i κi, (10)

where p_i is the inner-product column vector of positions between sensori and its nearby beacons, and κithe column

vector of ranging between them.

The corresponding coordinates of this sensor can be re-vealed by writing locally the expression (9), obtained from 1_{Different strategies can be proposed to define the neighborhood of a} sensor. This can be done by examining the ranging values, where high values correspond to neighbors. We fix their number in simulations.

for each sensor i

Find the nearby beacons [dump,ind]=sort(K(i,1:n))

Consider the closes nc beacons ind=ind(end:-1:end-nc+1)

Get ranging between these beacons Ki=K(ind,ind)

Get inner products between them Pi=Xn(ind,:)*Xn(ind,:)’

Consider ranging with them ki=K(i,ind)

Compute inner products with (10) pi=Pi*inv(Ki)*ki

Determine position with (11) y=pi/(Xn(ind,:)’) Table 1. Pseudocode of the distributed algorithm.

an eigen-decomposition problem. Then, a mapping trans-formation must be carried out as presented above, by con-sidering this time only neighboring beacons. While this be-comes fairly cumbersome for each sensor, we propose an alternative approach to find the coordinates, based on the pseudoinverse operator. For this purpose, we rewrite the problem as the following optimization problem

min

y kpi− Xiy ⊤_k2

F.

It is well known that the solution of this linear system is given by the left pseudoinverse of the matrix Xi, with

y= X⊤ i Xi

−1

X⊤i pi. (11)

We emphasize on the fact that we don’t need to apply a map-ping to localize the sensors with respect to beacons. The simplicity of the algorithm is illustrated in Table 1.

5. SIMULATIONS

To illustrate our method, we consider a configuration similar to the one proposed in [13], with ranging between two sen-sors is only a function of the distance between them, with

κ(xi, xj) = exp −kxi− xjk2/2σ2

, whereσ is a

param-eter corresponding to the range of the sensors. Next, sensors are randomly spread on a 1-by-1 square region.

0 5 10 15 20 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0 0.1 0.2 0.3 0.4 Numb erof beaco ns Ran_ge para mete_r R M S er ro r

Fig. 1. Root-mean-square error on positions with the

Fig. 2. Topology constructed by the centralized algorithm.

The beacons are represented by, sensors positions by+, and their estimations by◦.

In a first series of experiments, we apply the centralized algorithm to a network of 200 sensors, and study the in-fluence of both the number of beacons and the range pa-rameter. For this, we take n = 3, 4, . . . , 20 and σ = 0.1, 0.2, . . . , 0.9. In Fig. 1, we plot the resulting

root-mean-square error, averaged over 100 trials. As expected, the localization error decreases as the number of beacons increases, and the visibility between sensors is high. By taking for instance 15 beacons (almost 7% of the sensors), andσ = 0.75, we get the topology illustrated in Fig. 2.

In a second experiment, we consider a large scale network of 1000 sensors of low range, withσ = 0.3. We deploy

also 50 beacons with the same characteristics. This setting results in inverting and manipulating large sparse matrices, since there is low visibility between these entities. For these reasons, we consider the distributed algorithm, each sensor determines its coordinates with information from the 5 clos-est beacons. Fig. 3 illustrates the resulting topology, with a root-mean-square error of 0.018.

Fig. 3. Topology constructed by the distributed algorithm

for a large scale network (same legend as Fig. 2).

6. CONCLUSION

In this paper, we took advantage of recent works in ker-nel machines for solving the localization problem in sensor networks. We showed that the matrix regression method allows us to estimate unknown positions of sensors. We derived a distributed algorithm, based on information from local neighborhood of each sensor. There are several direc-tions for further research, including mobile ad hoc network (MANet), with an iterative update of the coordinates.

7. REFERENCES

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